Quantile Function

Given a random variable  with continuous and strictly monotonic probability density function , a quantile function  assigns to each probability  attained by  the value  for which . Symbolically,

Defining quantile functions for discrete rather than continuous distributions requires a bit more work since the discrete nature of such a distribution means that there may be gaps between values in the domain of the distribution function and/or "plateaus" in its range. Therefore, one often defines the associated quantile function  to be

where  denotes the range of .

REFERENCES:

Shaw, W. "Refinement of the Normal Quantile: A Benchmark Normal Quantile Based on Recursion, and an Appraisal of the Beasley-Springer-Moro, Acklam, and Wichura (AS241) Methods." 2007. https://www.mth.kcl.ac.uk/~shaww/web_page/papers/NormalQuantile1.pdf.